## Bernoulli

• Bernoulli
• Volume 25, Number 4A (2019), 2696-2728.

### Nonparametric Bayesian posterior contraction rates for scalar diffusions with high-frequency data

Kweku Abraham

#### Abstract

We consider inference in the scalar diffusion model $\,\mathrm{d}X_{t}=b(X_{t})\,\mathrm{d}t+\sigma(X_{t})\,\mathrm{d}W_{t}$ with discrete data $(X_{j\Delta_{n}})_{0\leq j\leq n}$, $n\to\infty$, $\Delta_{n}\to0$ and periodic coefficients. For $\sigma$ given, we prove a general theorem detailing conditions under which Bayesian posteriors will contract in $L^{2}$-distance around the true drift function $b_{0}$ at the frequentist minimax rate (up to logarithmic factors) over Besov smoothness classes. We exhibit natural nonparametric priors which satisfy our conditions. Our results show that the Bayesian method adapts both to an unknown sampling regime and to unknown smoothness.

#### Article information

Source
Bernoulli, Volume 25, Number 4A (2019), 2696-2728.

Dates
Revised: July 2018
First available in Project Euclid: 13 September 2019

https://projecteuclid.org/euclid.bj/1568362040

Digital Object Identifier
doi:10.3150/18-BEJ1067

Mathematical Reviews number (MathSciNet)
MR4003562

Zentralblatt MATH identifier
07110109

#### Citation

Abraham, Kweku. Nonparametric Bayesian posterior contraction rates for scalar diffusions with high-frequency data. Bernoulli 25 (2019), no. 4A, 2696--2728. doi:10.3150/18-BEJ1067. https://projecteuclid.org/euclid.bj/1568362040

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#### Supplemental materials

• Supplementary material for “Nonparametric Bayesian posterior contraction rates for scalar diffusions with high-frequency data.”. We supply proofs to accompany those in the article.